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arXiv:2107.12302 · Entropy

Quantum Heat Engines with Complex Working Media, Complete Otto Cycles and Heuristics

Ramandeep S. Johal, Venu Mehta

Published Sep 1, 2021• 14 citations
arXivHTMLPDF

Abstract

Quantum thermal machines make use of non-classical thermodynamic resources, one of which include interactions between elements of the quantum working medium. In this paper, we examine the performance of a quasi-static quantum Otto engine based on two spins of arbitrary magnitudes subject to an external magnetic field and coupled via an isotropic Heisenberg exchange interaction. It has been shown earlier that the said interaction provides an enhancement of cycle efficiency, with an upper bound that is tighter than the Carnot efficiency. However, the necessary conditions governing engine performance and the relevant upper bound for efficiency are unknown for the general case of arbitrary spin magnitudes. By analyzing extreme case scenarios, we formulate heuristics to infer the necessary conditions for an engine with uncoupled as well as coupled spin model. These conditions lead us to a connection between performance of quantum heat engines and the notion of majorization. Furthermore, the study of complete Otto cycles inherent in the average cycle also yields interesting insights into the average performance.

DOI
10.3390/e23091149

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Quantum thermal machines make use of non-classical thermodynamic resources, one of which include interactions between elements of the quantum working medium. In this paper, we examine the performance of a quasi-static quantum Otto engine based on two spins of arbitrary magnitudes subject to an external magnetic field and coupled via an isotropic Heisenberg exchange interaction. It has been shown earlier that the said interaction provides an enhancement of cycle efficiency, with an upper bound that is tighter than the Carnot efficiency. However, the necessary conditions governing engine performance and the relevant upper bound for efficiency are unknown for the general case of arbitrary spin magnitudes. By analyzing extreme case scenarios, we formulate heuristics to infer the necessary conditions for an engine with uncoupled as well as coupled spin model. These conditions lead us to a connection between performance of quantum heat engines and the notion of majorization. Furthermore, the study of complete Otto cycles inherent in the average cycle also yields interesting insights into the average performance.
The concept of majorization is explored as a tool to characterize the performance of a quantum Otto engine in the quasistatic regime. For a working substance in the form of a single spin of arbitrary magnitude, majorization yields a necessary and sufficient condition for the operation of the Otto engine, provided the canonical distribution of the working medium at the hot reservoir is majorized by its canonical distribution at the cold reservoir. We extend our analysis for two scenarios: a three-level atom and a bipartite system consisting of spin 1/2 interacting with an arbitrary spin via an isotropic Heisenberg exchange interaction. For both these cases, we find that while a majorization condition implies positive work extraction, it only yields sufficient conditions for the engine operation. Finally, we study the local thermodynamics of spins in the case of the bipartite system and infer an upper bound on the quantum Otto efficiency using the majorization relation.
We study the one-dimensional isotropic Heisenberg model of two spin-1/2 systems as a quantum heat engine. The engine undergoes a four-step Otto cycle where the two adiabatic branches involve changing the external magnetic field at a fixed value of the coupling constant. We find conditions for the engine efficiency to be higher than in the uncoupled model; in particular, we find an upper bound which is tighter than the Carnot bound. A domain of parameter values is pointed out which was not feasible in the interaction-free model. Locally, each spin seems to cause a flow of heat in a direction opposite to the global temperature gradient. This feature is explained by an analysis of the local effective temperature of the spins.
A properly selected quantum heat engine consumes resources efficiently. Also, there are many parameters to determine the right option. In this study, we construct and compare the quantum mechanical counterparts of Stirling and Otto cycles for an experimentally achievable quantum working medium where two spins 1/2 interact through isotropic Heisenberg interaction in the tunable external magnetic field. We consider two comparison cases that operate heat cycles with the same system parameters or equal efficiencies. The quantum Stirling cycle produces more work than that of the Otto cycle under the same parameters or equal efficiency conditions. The quantum Stirling engine operates a wider range of coupling strengths. The work performed in the cycles is enhanced by coupling strength. On the other hand, we investigate the role of coupling strength on the performance of the non-regenerative quantum Stirling heat cycle under various magnitudes of the magnetic field. The distance between the applied magnetic fields at the initial and final points of the isothermal stage is found to be an enhancer for the work output and efficiency of the quantum Stirling cycle.
We consider a quasistatic quantum Otto cycle using two effectively two-level systems with degeneracy in the excited state. The systems are coupled through isotropic exchange interaction of strength J>0, in the presence of an external magnetic field B which is varied during the cycle. We prove the positive work condition and show that level degeneracy can act as a thermodynamic resource, so that a larger amount of work can be extracted than in the nondegenerate case, both with and without coupling. We also derive an upper bound for the efficiency of the cycle. This bound is the same as derived for a system of coupled spin-1/2 particles [G. Thomas and R. S. Johal, Phys. Rev. E 83, 031135 (2011)PLEEE81539-375510.1103/PhysRevE.83.031135], i.e., without degeneracy, and depends only on the control parameters of the Hamiltonian, being independent of the level degeneracy and the reservoir temperatures.
We study quantum Otto thermal machines with a two-spin working system coupled by anisotropic interaction. Depending on the choice of different parameters, the quantum Otto cycle can function as different thermal machines, including a heat engine, refrigerator, accelerator, and heater. We aim to investigate how the anisotropy plays a fundamental role in the performance of the quantum Otto engine (QOE) operating in different timescales. We find that while the engine's efficiency increases with the increase in anisotropy for the quasistatic operation, quantum internal friction and incomplete thermalization degrade the performance in a finite-time cycle. Further, we study the quantum heat engine (QHE) with one of the spins (local spin) as the working system. We show that the efficiency of such an engine can surpass the standard quantum Otto limit, along with maximum power, thanks to the anisotropy. This can be attributed to quantum interference effects. We demonstrate that the enhanced performance of a local-spin QHE originates from the same interference effects, as in a measurement-based QOE for their finite-time operation.
Algebraic methods for solving time-dependent Hamiltonians are used to investigate the performance of quantum thermal machines. We investigate the thermodynamic properties of an engine formed by two coupled qubits, performing an Otto cycle. The thermal interaction occurs with two baths at different temperatures, while work is associated with the interaction with an arbitrary time-dependent magnetic field that varies in intensity and direction. For the coupling, we consider the 1D isotropic Heisenberg model, which allows us to describe the system by means of the irreducible representation of the su(2)\mathrm{su}(2)su(2) Lie algebra within the triplet subspace. We also inspect different settings of the temperatures and frequencies of the cycle, and investigate the corresponding operation regimes. We describe how these regimes can change with the speed of the protocol. Finally, we compare the weak and strong coupling regimes, and conclude that for the weak case the efficiency is optimal and can surpass the value of the classical Otto cycle. For low temperatures of the heat baths, we show that losses due to quantum friction are reduced in the strong coupling limit.